The Problem is: Let, $R_1$ and $R_2$ be two rings (with no mention of identity in either rings), then $R_1\times R_2$ forms a ring with component wise addition and multiplication. Then, give example of a ring $R = R_1\times R_2$ such that there exists an ideal $I$ of $R$ which is not of the form $I_1\times I_2$ for any $I_1$ and $I_2$ which are being ideals of $R_1$ and $R_2$ respectively .
My approach : I found a proof of a problem that if $R = R_1\times R_2$ contains the identity, then every ideal of $R$ is of the form $I_1\times I_2 .$ Rather, we know that every subgroup of a direct product of groups is not necessarily a direct product of subgroups of the component groups, then I was checking those subgroups were forming subrings or not by taking an example of the group $\mathbb Z_{14} \times \mathbb Z_7$, but it was not happening . Hence, I need hints to proceed further .
Bonus : What condition(s) is(are) to be imposed on $R$ such that every subring of $R$ is of the form $S_1\times S_2$ ???